Iwasawa Theory — Appendix A Duality in Galois cohomology

Appendix A
Duality in Galois cohomology

Fix a prime $p$ . For a field $E$ , let $G_{E}$ denote its absolute Galois group, i.e., the Galois group of its separable closure $E^{sep}$ as an extension of $E$ . Let $μ_{p^{\infty}}$ denote the group of all $p$ -power roots of unity in $E^{sep}$ .

Now let $E$ be a nonarchimedean local field of characteristic not equal to $p$ . Recall that its Brauer group $Br (E) = H^{2} (G_{E, S}, {(E^{sep})}^{\times})$ is canonically isomorphic to $ℚ_{p} ∕ ℤ_{p}$ by class field theory. This has the following corollary.

Lemma A.0.1. §

We have an isomorphism

H^{2} (G_{E}, ℚ_{p} ∕ ℤ_{p} (1)) \to_{}^{\sim} ℚ_{p} ∕ ℤ_{p} .

Proof.

Of course, we may replace $ℚ_{p} ∕ ℤ_{p} (1)$ by $μ_{p^{\infty}}$ in the statement, which in fact will make the isomorphism canonical. First, we remark that, since direct limits are exact, we have an isomorphism

H^{2} (G_{E}, μ_{p^{\infty}}) ≅ \underset{n}{lim_{\to}} H^{2} (G_{E}, μ_{p^{n}}) .

Kummer theory sets up an exact sequence

0 \to H^{2} (G_{E}, μ_{p^{n}}) \to Br (E) \to_{}^{p^{n}} Br (E),

the left exactness following from Hilbert’s theorem 90. Since the $p^{n}$ torsion in $Br (E)$ is canonically isomorphic to $1 ∕ p^{n} ℤ ∕ ℤ$ and $ℚ_{p} ∕ ℤ_{p}$ is the direct limit of the latter groups, we have the result. □

Remark A.0.2. §

If $T$ is a finite $ℤ_{p} [G_{E}]$ -module, then $H^{i} (G_{E}, T)$ is finite for every $i$ .

Theorem A.0.3 (Tate duality). §

Let $T$ be a finite $ℤ_{p} [G_{E}]$ -module. Then for $i \in ℤ$ the cup product

H^{i} (G_{E}, T) \times H^{2 - i} (G_{E}, T^{\lor} (1)) \to H^{2} (G_{E}, ℚ_{p} ∕ ℤ_{p} (1))

is nondegenerate, inducing an isomorphism

H^{i} (G_{E}, T) ≅ H^{2 - i} {(G_{E}, T^{\lor} (1))}^{\lor} .

Remark A.0.4. §

In fact, we have, more generally, such a duality for compact $ℤ_{p}$ -modules $T$ with continuous $G_{E}$ -actions. Here, we must use continuous cohomology, i.e., the cohomology groups of the complex of continuous $G_{E}$ -cochains with values in $T$ . We will in general denote such cohomology groups using the same notation as the usual profinite cohomology groups.

Finally, let $F$ denote a global field of characteristic not equal to $p$ , and let $S$ be a finite set of primes of $F$ .

Definition A.0.5. §

Let $T$ be a finite $ℤ_{p} [G_{F, S}]$ -module. For $i \in {1, 2}$ , the $i$ th Shafarevich-Tate group of $T$ is

Ш^{i} (G_{F, S}, T) = \ker (H^{i} (G_{F, S}, T) \to_{}^{\sum {Res}_{v}} ⨁_{v \in S} H^{i} (G_{F_{v}}, T)),

where the map ${Res}_{v}$ is the compostion of restriction to a decomposition group at $v \in S$ in $G_{F, S}$ with inflation to the absolute Galois $G_{F_{v}}$ .

The duality theorem is then as following

Theorem A.0.6 (Poitou-Tate duality). §

Let $T$ be a finite $ℤ_{p} [G_{F, S}]$ -module. For $i \in {1, 2}$ , we have isomorphisms

Ш^{i} (G_{F, S}, T) \to_{}^{\sim} Ш^{3 - i} {(G_{F, S}, T^{\lor} (1))}^{\lor} .

For $v \in S_{\infty}$ and any $i \in ℤ$ , we will use $H^{i} (G_{F_{v}}, T)$ to denote the $i$ th Tate cohomology group of $T$ , by abuse of notation.

Remark A.0.7. §

For $v \in S_{\infty}$ , the cup product induces isomorphisms

H^{i} (G_{F_{v}}, T) \to_{}^{\sim} H^{2 - i} {(G_{F_{v}}, T^{\lor} (1))}^{\lor}

for all $i \in ℤ$ .

Combining Poitou-Tate duality with Tate duality, we obtain the following nine-term exact sequence.

Theorem A.0.8 (Poitou-Tate sequence). §

For a finite $ℤ_{p} [G_{F, S}]$ -module $T$ , we have an exact sequence

The Poitou–Tate nine-term exact sequence. A full diagram description follows.

Diagram description: The Poitou–Tate nine-term exact sequence

This is one exact sequence, read left to right along the top row, then through the first dashed curved arrow to the middle row, then through the second dashed curved arrow to the bottom row, and finally to zero. The curved arrows start at the rightmost nonzero objects in their rows and end at the leftmost nonzero objects in the following rows. The nine cohomology terms, including the local direct sums and the dual Tate twists, are listed below.

Objects, listed by row and column:

Row 1, from left to right: column 2: 0; column 3: H superscript (0)(G subscript (F,S),T); column 4: direct sum subscript (v in S) H superscript (0)(G subscript (F subscript (v)),T); column 5: H superscript (2)(G subscript (F,S),T superscript (vee)(1)) superscript (vee).
Row 3, from left to right: column 3: H superscript (1)(G subscript (F,S),T); column 4: direct sum subscript (v in S) H superscript (1)(G subscript (F subscript (v)),T); column 5: H superscript (1)(G subscript (F,S),T superscript (vee)(1)) superscript (vee).
Row 5, from left to right: column 3: H superscript (2)(G subscript (F,S),T); column 4: direct sum subscript (v in S) H superscript (2)(G subscript (F subscript (v)),T); column 5: H superscript (0)(G subscript (F,S),T superscript (vee)(1)) superscript (vee); column 6: 0.

Arrows and lines:

An arrow from 0 (row 1, column 2) to H superscript (0)(G subscript (F,S),T), without a label.
An arrow from H superscript (0)(G subscript (F,S),T) to direct sum subscript (v in S) H superscript (0)(G subscript (F subscript (v)),T), without a label.
An arrow from direct sum subscript (v in S) H superscript (0)(G subscript (F subscript (v)),T) to H superscript (2)(G subscript (F,S),T superscript (vee)(1)) superscript (vee), without a label.
A dashed curved arrow from H superscript (2)(G subscript (F,S),T superscript (vee)(1)) superscript (vee) to H superscript (1)(G subscript (F,S),T), without a label.
An arrow from H superscript (1)(G subscript (F,S),T) to direct sum subscript (v in S) H superscript (1)(G subscript (F subscript (v)),T), without a label.
An arrow from direct sum subscript (v in S) H superscript (1)(G subscript (F subscript (v)),T) to H superscript (1)(G subscript (F,S),T superscript (vee)(1)) superscript (vee), without a label.
A dashed curved arrow from H superscript (1)(G subscript (F,S),T superscript (vee)(1)) superscript (vee) to H superscript (2)(G subscript (F,S),T), without a label.
An arrow from H superscript (2)(G subscript (F,S),T) to direct sum subscript (v in S) H superscript (2)(G subscript (F subscript (v)),T), without a label.
An arrow from direct sum subscript (v in S) H superscript (2)(G subscript (F subscript (v)),T) to H superscript (0)(G subscript (F,S),T superscript (vee)(1)) superscript (vee), without a label.
An arrow from H superscript (0)(G subscript (F,S),T superscript (vee)(1)) superscript (vee) to 0 (row 5, column 6), without a label.

Proof.

We first define the maps in question. The maps

{Res}_{v} : H^{i} (G_{F, S}, T) \to H^{i} (G_{F_{v}}, T)

are the compositions of the restriction maps from $G_{F, S}$ to a decomposition group above $v \in S$ with inflation to the absolute Galois group $G_{F_{v}}$ . The maps

H^{i} (G_{F_{v}}, T) \to H^{2 - i} {(G_{F, S}, T^{\lor} (1))}^{\lor}

are the compositions of the maps

H^{i} (G_{F_{v}}, T) \to H^{2 - i} {(G_{F_{v}}, T^{\lor} (1))}^{\lor}

of Tate duality with the Pontryagin duals of the maps ${Res}_{v}$ with for the module $T^{\lor} (1)$ . Finally the maps

H^{3 - i} {(G_{F, S}, T^{\lor} (1))}^{\lor} \to H^{1} (G_{F, S}, T)

are defined to be the natural maps that factor through the Poitou-Tate isomorphisms

Ш^{3 - i} {(G_{F, S}, T^{\lor} (1))}^{\lor} \to Ш^{i} (G_{F, S}, T) .

We briefly sketch the proof of exactness. Exactness at the first and last stages follows from injectivity of restriction on zeroth cohomology groups. Exactness at the local stages follows from global class field theory, which tells us that the image of $H^{i} (G_{F, S}, T)$ is the orthogonal complement of the image of $H^{2 - i} (G_{F, S}, T^{\lor} (1))$ under the sum of local cup products. (We omit the argument, but see [NSW, Section 8.6].) Finally, exactness at the other four global stages follows directly from Poitou-Tate duality. □

Remark A.0.9. §

As with Tate duality, we have Poitou-Tate duality and the Poitou-Tate sequence more generally for compact $ℤ_{p}$ -modules $T$ with continuous $G_{F, S}$ -actions.